Consider a Li$-$ion basery includes a graphite negative electrode and manganese$-$dioxide spinel prositive electrode, described by the following resetions $($discharge directoen$)$

$1.i_8{C}_{6}xL{i}^{+}+x{e}^{-}+C_6$

$xl.{?}^{+}+x{e}^{-}+M{n}_2{O}_{4}L{i}_{5}M{n}_2{O}_4$

The electrolyte consists of an organic solvent that contains a binary LIPF, lithium salt, where the anion is $P{F}_6.$ Assume a ID cell with the cathode located at $x=0$ and the anode locabed at $x=L.$ The properties are given as$,$

$D_{L^{+}}=1\mathrm{.}8{10}^{-14}\frac{m^2}{s}$

$D_{e{r}_4}=2\mathrm{.}6{10}^{-10}\frac{m^2}{s}$

$t^{+}=0\mathrm{.}4$

Fundamentals of Elichochewical Energy Sysfews

Coprigho $2024$: Parha P$.$ Mukherjec

Assume, electrodes are flat surfaces, and they are separated by an electrolyte$-$containing separator thst is $25m$ thick. The initial concentration of L$.$IPF in the electrobye is $1\mathrm{.}0$M$.$ The parameters given above are for trinsopert in the separater.

a$)$ State the direction of the current flow during discharge. Is it positive or negative direction relative to the $x$ direction?

b$)$ As only $L{i}^{+}$can transport the current, derive the following expression for Fithium$-$ion flux,

$N_{+}=-\frac{D}{1-t_{+}}\frac{dc}{dx}$

Where, $c$ is the salt coecentration.

c$)$ What is the exncentration difference across the separator if a cell with $22$cma$5cm$ electrode is operated at a current of $10$man $?$